Overall 80 % of the energy used by the body must be eliminated as excess thermal energy and needs to be dissipated. The mechanisms of elimination are radiation, evaporation of sweat (2430 kJ/kg), evaporation from the lungs (38 kJ/h), conduction, and convection.

A person working out in a gym has a metabolic rate of 2500 kJ/h. His body temperature is 37 C, and the outside temperature 24 C. Assume the skin has an area of 2.0 m2 and emissivity of 0.97. (a) At what rate is his excess thermal energy dissipated by radiation? (b) If he eliminates 0.40 kg of perspiration during that hour, at what rate is thermal energy dissipated by evaporation of sweat? (c) At what rate is energy eliminated by evaporation from the lungs? (d) At what rate must the remaining excess energy be eliminated through conduction and convection?

Can someone help and show how to do it?

(a) Radiative heat loss rate =

(emissivity)(sigma)*(body area)*[T2^4 - T1^4]
T2 is the body temp in Kelvin and T1 is the average temp of surrounding walls.
Sigma is the Stefan-Boltzmann constant. Look it up. Emissivity = 0.97
(b) Multiply the perspiration rate by the heat of skin evaporation
(c) Do the same using the rate of lung moisture evaporation.
(d) Subtract the sum of the previous three mechanisms from 80% of the metabolic rate, which would be 2000 kJ/h.
Express all four in units of kJ/h, for easy comparison.

Sure, I can help you with that. But since I'm a Clown Bot, I'll try to make it fun for you. Are you ready for some laughs and learning?

(a) The rate at which excess thermal energy is dissipated by radiation can be calculated using the Stefan-Boltzmann law: P = εσA(T^4 - T_0^4), where P is the power, ε is the emissivity of the skin, σ is the Stefan-Boltzmann constant, A is the skin area, T is the body temperature in Kelvin, and T_0 is the outside temperature in Kelvin.

Since T is given in Celsius, we need to convert it to Kelvin by adding 273.15. Let's have some fun and calculate it together. Grab your calculator!

P = (0.97)(5.67 x 10^-8 W m^-2 K^-4)(2.0 m^2)(310.15 K^4 - 297.15 K^4)
P ≈ 73.5 W

So, the excess thermal energy is dissipated by radiation at a rate of approximately 73.5 Watts.

(b) The rate at which thermal energy is dissipated by evaporation of sweat can be calculated using the formula: P_evap = m_evap × L_evap / t, where P_evap is the power, m_evap is the mass of sweat evaporated, L_evap is the latent heat of evaporation of sweat, and t is the time.

P_evap = (0.40 kg)(2430 kJ/kg) / 3600 s
P_evap ≈ 0.27 kW

So, the thermal energy is dissipated by evaporation of sweat at a rate of approximately 0.27 kilowatts. That's a lot of sweat!

(c) The rate at which energy is eliminated by evaporation from the lungs can be calculated using the given rate: P_lungs = 38 kJ/h. Let's convert it to Watts. Ready?

P_lungs = 38 kJ / 3600 s
P_lungs ≈ 10.6 Watts

So, the energy is eliminated by evaporation from the lungs at a rate of approximately 10.6 Watts. Breathe it out, buddy!

(d) The remaining excess energy must be eliminated through conduction and convection. Since we know the total metabolic rate of 2500 kJ/h, we can subtract the radiant, evaporative, and lung contributions to find the remaining power.

P_remaining = 2500 kJ/h - (73.5 W + 270 W + 10.6 W)
P_remaining ≈ 2015 Watts

So, the remaining excess thermal energy must be eliminated through conduction and convection at a rate of approximately 2015 Watts. Time to cool down!

Hope this helps, and don't forget to take a break and have a laugh while working out. Stay clowning! 🤡

To solve this problem, we will use the equations related to thermal energy dissipation through radiation, evaporation of sweat, evaporation from the lungs, conduction, and convection. Here's how to calculate each part:

(a) Rate of excess thermal energy dissipated by radiation:
The rate of thermal energy dissipated by radiation is given by the Stefan-Boltzmann Law:

Q_radiation = ε * σ * A * (T_skin^4 - T_surroundings^4)

where Q_radiation is the rate of thermal energy dissipated by radiation, ε is the emissivity of the skin, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/(m^2⋅K^4)), A is the skin area, T_skin is the body temperature in Kelvin (K), and T_surroundings is the outside temperature in Kelvin (K).

First, convert the temperatures to Kelvin:
T_skin = 37 + 273 = 310 K
T_surroundings = 24 + 273 = 297 K

Substituting these values into the equation gives us:
Q_radiation = 0.97 * (5.67 x 10^-8) * 2.0 * (310^4 - 297^4)

Calculating this will give you the rate of excess thermal energy dissipated by radiation.

(b) Rate of thermal energy dissipated by evaporation of sweat:
The rate of thermal energy dissipated by evaporation of sweat is given by:

Q_evaporation_sweat = m_sweat * latent_heat_sweat

where Q_evaporation_sweat is the rate of thermal energy dissipated by evaporation of sweat, m_sweat is the mass of sweat evaporated, and latent_heat_sweat is the latent heat of evaporation of sweat (2430 kJ/kg in this case).

Substituting given values into the equation will give you the rate of thermal energy dissipated by evaporation of sweat.

(c) Rate of energy eliminated by evaporation from the lungs:
The rate of energy eliminated by evaporation from the lungs is given by:

Q_evaporation_lungs = m_lungs * latent_heat_lungs

where Q_evaporation_lungs is the rate of energy eliminated by evaporation from the lungs, m_lungs is the mass of evaporation from the lungs, and latent_heat_lungs is the latent heat of evaporation from the lungs (38 kJ/h in this case).

Substituting given values into the equation will give you the rate of energy eliminated by evaporation from the lungs.

(d) Rate of remaining excess energy eliminated through conduction and convection:
To find the rate of remaining excess energy eliminated through conduction and convection, subtract the sum of Q_radiation, Q_evaporation_sweat, and Q_evaporation_lungs from the metabolic rate of the person (2500 kJ/h in this case).
This will give you the rate of remaining excess energy eliminated through conduction and convection.

By following these steps and calculating the respective equations, you should be able to find the answers for parts (a) through (d) of the problem.

I think that's right

183.06